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(SOS)<\/jats:italic> has the form \n<jats:inline-formula><jats:alternatives><jats:tex-math>$$f = \\sum_{i\\in [s]} c_i f_i^2$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mrow>\n                    <mml:mi>f<\/mml:mi>\n                    <mml:mo>=<\/mml:mo>\n                    <mml:msub>\n                      <mml:mo>\u2211<\/mml:mo>\n                      <mml:mrow>\n                        <mml:mi>i<\/mml:mi>\n                        <mml:mo>\u2208<\/mml:mo>\n                        <mml:mo>[<\/mml:mo>\n                        <mml:mi>s<\/mml:mi>\n                        <mml:mo>]<\/mml:mo>\n                      <\/mml:mrow>\n                    <\/mml:msub>\n                    <mml:msub>\n                      <mml:mi>c<\/mml:mi>\n                      <mml:mi>i<\/mml:mi>\n                    <\/mml:msub>\n                    <mml:msubsup>\n                      <mml:mi>f<\/mml:mi>\n                      <mml:mi>i<\/mml:mi>\n                      <mml:mn>2<\/mml:mn>\n                    <\/mml:msubsup>\n                  <\/mml:mrow>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula>, where the <jats:italic>weights<\/jats:italic><jats:inline-formula><jats:alternatives><jats:tex-math>$$c_i$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:msub>\n                    <mml:mi>c<\/mml:mi>\n                    <mml:mi>i<\/mml:mi>\n                  <\/mml:msub>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula> are field elements. \nThe size of the representation is the number of monomials that appear across the <jats:inline-formula><jats:alternatives><jats:tex-math>$$f_i$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:msub>\n                    <mml:mi>f<\/mml:mi>\n                    <mml:mi>i<\/mml:mi>\n                  <\/mml:msub>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula>'s.\nIts minimum across all such decompositions is called the <jats:italic>support-sum S(f)<\/jats:italic> of <jats:italic>f<\/jats:italic>.<\/jats:p><jats:p>For a univariate polynomial <jats:italic>f<\/jats:italic> of degree <jats:italic>d<\/jats:italic> of full support,\na lower bound for the support-sum  is <jats:inline-formula><jats:alternatives><jats:tex-math>$$S(f) \\ge \\sqrt d$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mrow>\n                    <mml:mi>S<\/mml:mi>\n                    <mml:mrow>\n                      <mml:mo>(<\/mml:mo>\n                      <mml:mi>f<\/mml:mi>\n                      <mml:mo>)<\/mml:mo>\n                    <\/mml:mrow>\n                    <mml:mo>\u2265<\/mml:mo>\n                    <mml:msqrt>\n                      <mml:mi>d<\/mml:mi>\n                    <\/mml:msqrt>\n                  <\/mml:mrow>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula>.\nWe show that the existence of an explicit univariate polynomial <jats:italic>f<\/jats:italic>\nwith support-sum just slightly larger than the lower bound, that is, <jats:inline-formula><jats:alternatives><jats:tex-math>$$S(f) \\ge d^{0.5+\\varepsilon}$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mrow>\n                    <mml:mi>S<\/mml:mi>\n                    <mml:mrow>\n                      <mml:mo>(<\/mml:mo>\n                      <mml:mi>f<\/mml:mi>\n                      <mml:mo>)<\/mml:mo>\n                    <\/mml:mrow>\n                    <mml:mo>\u2265<\/mml:mo>\n                    <mml:msup>\n                      <mml:mi>d<\/mml:mi>\n                      <mml:mrow>\n                        <mml:mn>0.5<\/mml:mn>\n                        <mml:mo>+<\/mml:mo>\n                        <mml:mi>\u03b5<\/mml:mi>\n                      <\/mml:mrow>\n                    <\/mml:msup>\n                  <\/mml:mrow>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula>, \nfor some <jats:inline-formula><jats:alternatives><jats:tex-math>$$\\varepsilon &gt; 0$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mrow>\n                    <mml:mi>\u03b5<\/mml:mi>\n                    <mml:mo>&gt;<\/mml:mo>\n                    <mml:mn>0<\/mml:mn>\n                  <\/mml:mrow>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula>,\nimplies that <jats:inline-formula><jats:alternatives><jats:tex-math>$$\\ne$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mo>\u2260<\/mml:mo>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula>,\n<jats:italic>the<\/jats:italic> major open problem in algebraic complexity.\nIn fact, our proof works for some subconstant functions <jats:inline-formula><jats:alternatives><jats:tex-math>$$\\varepsilon(d) &gt; 0$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mrow>\n                    <mml:mi>\u03b5<\/mml:mi>\n                    <mml:mo>(<\/mml:mo>\n                    <mml:mi>d<\/mml:mi>\n                    <mml:mo>)<\/mml:mo>\n                    <mml:mo>&gt;<\/mml:mo>\n                    <mml:mn>0<\/mml:mn>\n                  <\/mml:mrow>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula> as well.\nWe also consider the <jats:italic>sum-of-cubes representation (SOC)<\/jats:italic> of polynomials. 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